We can thus embed them all in the same weighted projective space. Then one can remove
those suborbifolds of weighted projective space whose embedding is non-normal,
since they are easily seen to be unstable for the action of \(\prod _iGL(V^{k+i})\) . Thus by
the above result, orbits on the Hilbert scheme really corresponds to isomorphism classes
of polarised orbifolds.
Finally one should compactify with orbischemes (or Deligne-Mumford stacks)
to get proper moduli spaces of stable objects. We do not pursue this here as only smooth
orbifolds and their stability are relevant to cscK metrics, but many of the foundations
are worked out in [1]}. (Their point of view is slightly
different from ours – their notion of stability is related to the minimal model programme
rather than GIT, and they form moduli using the machinery of stacks.)
